To find the length of the third side of a triangle when given two sides and the included angle, we use the Law of Cosines. The Law of Cosines formula is:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cos(C) \][/tex]
Here, we are given:
- side [tex]\( a = 2 \)[/tex]
- side [tex]\( b = 5 \)[/tex]
- the angle between these sides [tex]\( C = 60^{\circ} \)[/tex]
Step 1: Convert the angle from degrees to radians. We know that [tex]\(60^{\circ}\)[/tex] is equivalent to [tex]\(\frac{\pi}{3}\)[/tex] radians.
Step 2: Apply the Law of Cosines formula. Substitute the values into the formula:
[tex]\[ c^2 = 2^2 + 5^2 - 2 \cdot 2 \cdot 5 \cdot \cos(60^{\circ}) \][/tex]
Step 3: Calculate the cosine of [tex]\( 60^\circ \)[/tex]. We know [tex]\(\cos(60^\circ) = \frac{1}{2} \)[/tex].
Step 4: Substitute [tex]\(\cos(60^\circ)\)[/tex] back into the equation:
[tex]\[ c^2 = 2^2 + 5^2 - 2 \cdot 2 \cdot 5 \cdot \frac{1}{2} \][/tex]
Step 5: Simplify the equation step by step:
[tex]\[ c^2 = 4 + 25 - 2 \cdot 2 \cdot 5 \cdot \frac{1}{2} \][/tex]
[tex]\[ c^2 = 4 + 25 - 10 \][/tex]
[tex]\[ c^2 = 4 + 25 - 10 = 19 \][/tex]
So,
[tex]\[ c^2 = 19 \][/tex]
Step 6: Take the square root of both sides to find [tex]\( c \)[/tex]:
[tex]\[ c = \sqrt{19} \][/tex]
Thus, the length of the third side is [tex]\(\sqrt{19}\)[/tex].
The correct answer is:
B. [tex]\(\sqrt{19}\)[/tex]
